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Morphing is the process of changing a geometric model or an image into another. The process generally involves rigid body motions and non-rigid deformations. It is well known that there exists a unique conformal mapping from a simply connected surface into a unit disk by the Riemann mapping theorem. On the other hand, a 3D surface deformable model can be built via various approaches such as mutual parameterization from direct interpolation or surface matching using landmarks. In this paper, a numerical methods of 3D surface morphing based on deformable model and conformal mapping is demonstrated. We take the advantage of the unique representation of 3D surfaces by the mean curvatures $H$ and the conformal factors $\lambda$ associated with the Riemann mapping and build up the deformation model by consistently registering the landmarks on the conformal parametric domains. As a result, the correspondence of the $(H, \lambda)$ between two surfaces can be defined and a 3D deformation field can be reconstructed. Furthermore, by composition of the M\"obius transformation and the 3D deformation field, a smooth morphing sequence can be generated over a consistent mesh structure via the cubic spline homotopy. Several numerical experiments on the face morphing are presented to demonstrate the robustness of our approach.

It is well known that the nonlinear filter has important applications in military, engineering and commercial industries. In this paper, we propose efficient and accurate numerical algorithms for the realization of the Yau-Yau method for solving nonlinear filtering problems by using finite difference schemes. The Yau-Yau method reduces the nonlinear filtering problem to the initial-value problem of Kolmogorov equations. We first solve this problem by the implicit Euler method, which is stable in most cases, but costly. Then, we propose a quasi-implicit Euler method which is feasible for acceleration by fast Fourier transformations. Furthermore, we propose a superposition technique which enables us to deal with the nonlinear filtering problem in an off-time process and thus, save a large amount of computational cost. Next, we prove that the numerical solutions of Kolmogorov equations by our schemes are always nonnegative in each iteration. Consequently, our iterative process preserves the probability density functions. In addition, we prove convergence of our schemes under some mild conditions. Numerical results show that the proposed algorithms are efficient and promising.

In this paper, a brief introduction of the nonlinear filtering problems and a review of the quasi-implicit Euler method are presented. The major contribution of this paper is that we propose a nonnegativity-preserving algorithm of Yau-Yau method for solving high-dimensional nonlinear filtering problems by applying quasi-implicit Euler method with discrete sine transform. Furthermore, our algorithms are directly applicable on the compact difference schemes, so that the number of spatial points can be substantially reduced and retain the same accuracy. Numerical results indicate that the proposed algorithm is capable of solving up to six-dimensional nonlinear filtering problems efficiently and accurately.

Surface parameterizations have been widely applied to digital geometry processing. In this paper, we propose an efficient conformal energy minimization (CEM) algorithm for computing conformal parameterizations of simply-connected open surfaces with a very small angular distortion and a highly improved computational efficiency. In addition, we generalize the proposed CEM algorithm to computing conformal parameterizations of multiply-connected surfaces. Furthermore, we prove the existence of a nontrivial accumulation point of the proposed CEM algorithm under some mild conditions. Several numerical results show the efficiency and robustness of the CEM algorithm comparing to the existing state-of-the-art algorithms. An application of the CEM on the surface morphing between simply-connected open surfaces is demonstrated thereafter. Thanks to the CEM algorithm, the whole computations for the surface morphing can be performed efficiently and robustly.

Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with very small area distortions and highly improved computational efficiencies. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the accuracy, effectiveness, and robustness of the proposed SEM algorithm outperform the other state-of-the-art algorithms. Applications of the SEM on surface remeshing, registration and morphing for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computation for these applications can be carried out efficiently and reliably.

Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle- and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.

Surface parameterizations have been widely applied in the computer-aided design for the geometric processing tasks of surface registration, remeshing, texture mapping, and so on. In this paper, we present an efficient balanced energy minimization (BEM) algorithm for the computation of simply connected open surface parameterizations with balanced angle and area distortions. The existence of a nontrivial accumulation function of the BEM algorithm is guaranteed under some mild conditions and the limiting function is shown to be one-to-one. Comparisons of the BEM algorithm with the angle- and the area-preserving parameterizations show that the angular distortion is close to that of the angle-preserving parameterization while the area distortion is significantly improved. An application of the BEM on the Chinese virtual broadcasting technique is demonstrated thereafter, which is consisted of surface remeshing, registration, and morphing.

Parameterizations of manifolds are widely applied to the fields of numerical partial differential equations and computer graphics. To this end, in recent years several efficient and reliable numerical algorithms have been developed by different research groups for the computation of triangular and tetrahedral mesh parameterizations. However, it is still challenging when the topology of manifolds is nontrivial, e.g., the 3-manifold of a topological solid torus. In this paper, we propose a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of toroidal polyhedra with a single boundary being mapped to a standard torus. In addition, the algorithm can also be used to compute the equiareal mapping between a genus-one closed surface and the standard torus. Numerical experiments indicate that the developed algorithm is effective and performs well on the bijectivity of the mapping. Applications on manifold registrations and partitions are demonstrated to show the robustness of our algorithms.

Optimal mass transportation has been widely applied in various fields, such as data compression, generative adversarial networks, and image processing. In this paper, we adopt the projected gradient method, combined with the homotopy technique, to nd a minimal volume-measure-preserving solution for a 3-manifold optimal mass transportation problem. The proposed projected gradient method is shown to be sublinearly convergent at a rate of O(1/k). Several numerical experiments indicate that our algorithms are able to signicantly reduce transportation costs. Some applications of the optimal mass transportation maps - to deformations and canonical normalizations between brains and solid balls - are demonstrated to show the robustness of our proposed algorithms.

Optimal mass transport (OMT) theory is used to preprocess brain tumor datasets for the first time. The goal is to move any irregular 3D object (i.e., the brain) without causing significant distortion. The first stage of a two-stage OMT (TSOMT) procedure transforms the brain onto the unit solid ball. The sphere shape boundary constraint is necessary to ensure OMT convergence. The second stage is to transform the unit ball to a unit cube, as it is easier to apply a 3D convolutional neural network (CNN) to rectangular coordinates. Small variations in the local mass-measured stretch ratio among all the brain tumor datasets show the robustness of the transform. Additionally, the distortion is kept at a minimum (<0.1) with a reasonable transport cost. The original 240 × 240 × 155 × 4 dataset is thus reduced to a cube of 128 × 128 × 128 × 4, which is a 76.6% reduction in the total number of voxels, without losing much detail. Two mass-preserving OMT functions (one is fluid-attenuated inversion recovery (FLAIR) only and the other is 4 modalities on average) are used to double the amount of training data, which helps to reduce overfitting when training a vast machine learning (ML) model. Three typical U-Nets are trained separately to predict the whole tumor (WT), tumor core (TC), and contrast-enhancing tumor (CET) from the cube. An impressive 0.9901 training accuracy in the WT cube is achieved at 1000 epochs. Inverse TSOMT is performed on the predicted cube so that brain results are obtained. The conversion loss from OMT to inverse OMT is found to be less than one percent. For training, the Dice scores (0.9852 for the WT, 0.9743 for the TC, and 0.9433 for the CET) can be obtained. Significant improvements in brain tumor detection and the segmentation accuracy have been achieved. For validation, postprocessing using rotating techniques is added to the TSOMT, U-Net prediction, and inverse TSOMT method for an accuracy improvement of a few percent. It takes 200 seconds to complete the whole segmentation process on each new brain tumor dataset. Finally, the false prediction for the worst WT, TC and CET validation cases is discussed. Again, the TSOMT and inverse TSOMT do not induce any new failure mode because of the small conversion loss.